General Formulation of an Efficient Recursive Algorithm Based on Canonical Momenta for Forward Dynamics of Closed-Loop Multibody Systems

Abstract

In previous work, a method for establishing the equations of motion of open-loop multibody mechanisms was introduced. The proposed forward dynamics formulation resulted in a Hamiltonian set of 2n first order ODE’s in the generalized coordinates q and the canonical momenta p. These Hamiltonian equations were derived from a recursive Newton-Euler formulation. It was shown how an O(n) formulation could be obtained in the case of a serial structure with general joints. The amount of required arithmetical operations was considerably less than comparable acceleration based formulations. In this paper, a further step is taken: the method is extended to constrained multibody systems. Using the principle of virtual power, it is possible to obtain a recursive Hamiltonian formulation for closed-loop mechanisms as well, enabling the combination of the low amount of arithmetical operations and a better evolution of the constraints violation errors, when compared with acceleration based methods.

O(n) mass matrix inversion for serial manipulators and polypeptide chains using Lie derivatives

“…Particularly, if we define the SE(3) gradient of a function to be then we can apply this gradient to the parameters q = [q 1 , …, q 6 ] T used to parameterize a motion. Using (17) and observing that , we find that (18) This means that the inverse of the mass matrix for a single rigid-body can be written in the Fixman-like form: (19) For a collection of n rigid bodies, the configuration space is (SE (3)…”

“…Hence, it differs from the direct sum of the inverse of Jacobian matrices, , where denotes the inverse Jacobian of the i th rigid-body. The inverse of the unconstrained mass matrix for this collection of rigid bodies is then of the form (20) Everything then follows using the extension of Fixman's theorem as in the point-mass case, with (20) replacing (4). The same partitioning into soft and hard variables and the same O(n) performance results.…”

“…A coordinate invariant algorithm for forward dynamics using Lie groups and Lie algebra of SE (3) was introduced in [18]. More recently, a recursive O(n) forward dynamic computations was used to obtain a set of Hamiltonian equations for open-loop and closed-loop multi-body systems [19,20].…”

Recursive Algorithm Based on Canonical Momenta for Forward Dynamics of Multibody Systems: Numerical Results

“…A new, canonical momenta based algorithm was presented to solve the forward dynamics problem in a very efficient way, by reducing the number of operations required to obtain the equations of motion. It is also possible to handle constrained multibody systems with the algorithm, as shown in [15]. The purpose of this paper is to demonstrate some numerical examples and to point out some interesting features of the Hamiltonian equations related to the numerical integrations of these specific examples.…”

“…In previous publications [13,14], an attempt was made to address the first challenge. A new, canonical momenta based algorithm was presented to solve the forward dynamics problem in a very efficient way, by reducing the number of operations required to obtain the equations of motion.…”